The most important examples of sparse codes are topological stabilizer
codes, also called homology codes because of their relation to
homology theory. The archetype
for this code family is Kitaev's toric code [1]}, which encodes \(k=O(1)\)  qubits
and has minimum distance \(d=O(\sqrt{n})\)  (although it can correct a constant fraction of
random errors). It is known that 2\(D\)  homological codes obey
\(d \le O(\sqrt{n})\)  [2]}. Many other important examples of such codes are
known; see Ref. [3]} for a survey.
